Flexural Vibrations of Thin Rods. 
395 
1908-9.] 
hydrodynamical potential function, with x and t interchanged, is a solution 
of equation (3) above. (The converse is not always true.) In relation to 
the flexural waves problems £ is in general to be treated as a constant. 
§ 4. It may be of some interest to note that, in all solutions for elastic 
bars obtained by applying the above result, the displacement y at each 
point of the bar may be regarded as derived from a single function 
Y(t, z, x) such that 
V 
0Y 
dx 
( 10 ). 
Further, it follows from the relationship established between such a function 
V and the hydrodynamic potential function, that every function derived 
from V by differentiations or integrations with respect to t, z , x, is a solu- 
tion of the differential equation (3). Thus any solution of (3) may be a 
displacement potential, or a displacement, or a velocity. If the function 
V represents displacement, we readily find from equation (7) that the 
curvature at each point x of the bar can be obtained by a single differentia- 
tion with respect to z, being given by the equation 
i _ i ay 
R kI) dz 
(II), 
from which the potential energy can easily be obtained. 
§ 5. Solutions for vibrations of a rod of finite length l are derived 
directly from the hydrodynamic solutions for waves in a canal of length l 
by simple interchange of x and t ; and they are applicable to all cases, 
whether the rod has its ends free, or clamped, or “ supported.” In 
particular, the normal functions are easily obtained in this way. 
In the case of an infinite rod, all mathematical results relating to surface 
waves in a canal infinitely long and infinitely deep become immediately 
useful ; space-curves in the hydrodynamical waves problems becoming 
time-curves for the flexural waves, and vice versa. 
§ 6. In this connection it may be useful at a later time to examine some 
of the numerous hydrodynamical solutions relating to surface waves and 
groups of waves. A number of curves are shown in papers on Water- 
Waves * by the late Lord Kelvin, illustrating results derived from particular 
hydrodynamic solutions comprehended in the following general expression, 
given in his last Waves paper : — 
d j+k+l l - ,° vl - x 
{RS}or { RD}^^^ )e 
In this, {RS} denotes a realisation by taking half the sum of what follows 
* Proc. Poy. Soc. Edin vol. xxv., Feb. and June 1904 ; vol. xxvi., Oct. 1906. 
